some results of algebraic geometry over henselian rank one … · 2017-04-11 · theory of stably...

Sel. Math. New Ser. (2017) 23:455–495DOI 10.1007/s00029-016-0245-y

Selecta MathematicaNew Series

Some results of algebraic geometry over Henselian rankone valued fields

Krzysztof Jan Nowak1

Published online: 13 June 2016© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract Wedevelop geometry of affine algebraic varieties in K n overHenselian rankone valued fields K of equicharacteristic zero. Several results are provided including:the projection K n × P

m(K ) → K n and blowups of the K -rational points of smoothK -varieties are definably closed maps; a descent property for blowups; curve selec-tion for definable sets; a general version of the Łojasiewicz inequality for continuousdefinable functions on subsets locally closed in the K -topology; and extending con-tinuous hereditarily rational functions, established for the real and p-adic varieties inour joint paper with J. Kollár. The descent property enables application of resolutionof singularities and transformation to a normal crossing by blowing up in much thesame way as over the locally compact ground field. Our approach relies on quantifierelimination due to Pas and a concept of fiber shrinking for definable sets, which is arelaxed version of curve selection. The last three sections are devoted to the theory ofregulous functions and sets over such valued fields. Regulous geometry over the realground field R was developed by Fichou–Huisman–Mangolte–Monnier. The mainresults here are regulous versions of Nullstellensatz and Cartan’s theorems A and B.

Keywords Closedness theorem · Descent property for blowups · Curve selection ·Łojasiewicz inequality · Hereditarily rational functions · Regulous functions andsets · Nullstellensatz · Cartan’s theorems A and B

Mathematics Subject Classification Primary 12J25 · 03C10; Secondary 14G27 ·14P10

B Krzysztof Jan [email protected]

1 Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University,ul. Profesora Łojasiewicza 6, 30-348 Kraków, Poland

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456 K. J. Nowak

1 Introduction

In this paper, we develop geometry of affine algebraic varieties in K n over Henselianrank one valued fields K of equicharacteristic zero with valuation v, value group �,valuation ring R and residue field k. Every rank one valued field has a metric topologyinduced by its absolute value. Examples of such fields are the quotient fields of therings of formal power series and of Puiseux series with coefficients from a field k ofcharacteristic zero as well as the fields of Hahn series (maximally complete valuedfields also called Malcev–Neumann fields; cf. [24]):

k((t�)) :=⎧⎨

⎩f (t) =

∑

γ∈�

aγ tγ : aγ ∈ k, supp f (t) is well ordered

⎫⎬

⎭.

Let X be a K -algebraic variety. We always assume that X is reduced, but we allowit to be reducible. The set X (K ) of its K -rational points (K -points for short) inheritsfrom K a topology, called the K -topology. In this paper, we are going to investigatecontinuous and differentiable functions X (K ) → K that come from algebraic geom-etry and their zero sets. Therefore, we shall (and may) most often assume that X isan affine K -variety such that X (K ) is Zariski dense in X . Throughout the paper, by“definable” we shall mean “definable with parameters.”

Several results concerning algebraic geometry over such ground fields K are estab-lished. Let L be the three-sorted language of Denef–Pas. We prove that the projection

K n × Pm(K ) → K n

is an L-definably closed map (Theorem 3.1). Further, we shall draw several conclu-sions, including the theorem that blowups of the K -points of smooth K -varieties aredefinably closed maps (Corollary 3.5), a descent property for such blowups (Corol-lary 3.6), curve selection for L-definable sets (Proposition 8.2) and for valuativesemialgebraic sets (Proposition 8.1) as well as a general version of the Łojasiewiczinequality for continuous L-definable functions on subsets locally closed in theK -topology (Proposition 9.2). Also given is a theorem on extending continuous hered-itarily rational functions over such ground fields (Theorem 10.2), established for thereal and p-adic varieties in our joint paper [27] with J. Kollár. The proof makes useof the descent property and the Łojasiewicz inequality. The descent property enablesapplication of resolution of singularities and transformation to a normal crossing byblowing up (see [25, Chap. III] for references and relatively short proofs) in much thesame way as over the locally compact ground field. Our approach relies on quantifierelimination due to Pas and on a certain concept of fiber shrinking for definable sets,which is a relaxed version of curve selection. Note that this paper comprises our twoearlier preprints [39,40].

Remark 1.1 This paper is principally devoted to geometry over rank one valued fields(in other words, fields with non-archimedean absolute value). Therefore, from Sect. 3on, we shall most often assume that so is the ground field K . Nevertheless, it is plau-sible that the closedness theorem (Theorem 2.6) and curve selection (Propositions 8.1and 8.2) hold over arbitrary Henselian valued fields.

Some results of algebraic geometry over Henselian… 457

We should emphasize that our approach to the subject of this paper is possiblejust because the language L in which we investigate valued fields is not too rich; inparticular, it does not contain the inclusion language on the auxiliary sorts and theonly symbols of L connecting the sorts are two functions from the main K -sort tothe auxiliary �-sort and k-sort. Hence and by elimination of K -quantifiers, the L-definable subsets of the products of the two auxiliary sorts are precisely finite unionsof the Cartesian products of sets definable in those two sorts. This allows us to reduceour reasonings to an analysis of ordinary cells (i.e., fibers of a cell in the sense of Pas).

The organization of the paper is as follows. In Sect. 2, we set up notation andterminology including, in particular, the language L of Denef–Pas and the conceptof a cell. We recall the theorems on quantifier elimination and on preparation celldecomposition, due to Pas [41]. Next we draw some conclusions as, for instance,Corollary 2.3 on definable functions and Corollary 2.7 on certain decompositions ofdefinable sets. The former will be applied in Sect. 5, and the latter is crucial for ourproof of the closedness theorem (Theorem 3.1), which is stated in Sect. 3 togetherwith several direct corollaries, including the descent property. Section 4 gives a proof(being of algorithmic character) of this theorem for the case where the value group �

is discrete.In Sect. 5, we study L-definable functions of one variable. A result playing an

important role in the sequel is the theorem on existence of the limit (Proposition 5.2).Its proof makes use of Puiseux’s theorem for the local ring of convergent powerseries. In Sect. 6, we introduce a certain concept of fiber shrinking forL-definable sets(Proposition 6.1), which is a relaxed version of curve selection. Section 7 provides aproof of the closedness theorem (Theorem 3.1) for the general case. This proof makesuse of fiber shrinking and existence of the limit for functions of one variable.

In the subsequent three sections, some further conclusions from the closednesstheorems are drawn. Section 8 provides some versions of curve selection: for arbitraryL-definable sets and for valuative semialgebraic sets. The next section is devoted to ageneral version of the Łojasiewicz inequality for continuous L-definable functions onsubsets locally closed in the K -topology (Proposition 9.2). In Sect. 10, the theorem onextending continuous hereditarily rational functions (established for the real and p-adic varieties in [27]) is carried over to the casewhere the groundfield K is aHenseliamrank one valued field of equicharacteristic zero. Let us mention that in real algebraicgeometry applications of continuous hereditarily rational functions and the extensiontheorem, in particular, are given in the papers [28–30] and [31], which discuss rationalmaps into spheres and stratified-algebraic vector bundles on real algebraic varieties.

The last three sections are devoted to the theory of regulous functions and sets overHenselian rank one valued fields of equicharacteristic zero. Regulous geometry overthe real ground field R was developed by Fichou–Huisman–Mangolte–Monnier [16].In Sect. 11, we set up notation and terminology as well as provide basic results aboutregulous functions and sets, including the noetherianity of the constructible and regu-lous topologies. Those results are valid over arbitrary fields with the density property.The next section establishes a regulous version of Nullstellensatz (Theorem 12.4),valid over Henselian rank one valued fields of equicharacteristic zero. The proof relieson the Łojasiewicz inequality (Proposition 9.2). Also drawn are several conclusions,including the existence of a one-to-one correspondence between the radical ideals of

458 K. J. Nowak

the ring of regulous functions and the closed regulous subsets, or one-to-one corre-spondences between the prime ideals of that ring, the irreducible regulous subsets andthe irreducible Zariski closed subsets (Corollaries 12.5 and 12.10).

Section 13 provides an exposition of the theory of quasi-coherent regulous sheaves,which generally follows the approach given in the real algebraic case by Fichou–Huisman–Mangolte–Monnier [16]. It is based on the equivalence of categoriesbetween the category of Rk-modules on the affine scheme Spec

(Rk(K n))and the cat-

egory ofRk-modules on K n which, in turn, is a direct consequence of the one-to-onecorrespondences mentioned above. The main results here are the regulous versions ofCartan’s theorems A and B. We also establish a criterion for a continuous function onan affine regulous subvariety to be regulous (Proposition 13.10), which relies on ourtheorem on extending continuous hereditarily rational functions (Theorem 10.2).

Note finally that the metric topology of a non-archimedean field K with a rankone valuation v is totally disconnected. Rigid analytic geometry (see, e.g., [6] for itscomprehensive foundations), developed by Tate, compensates for this defect by intro-ducing sheaves of analytic functions in a Grothendieck topology. Another approachis due to Berkovich [3], who filled in the gaps between the points of K n , producinga locally compact Hausdorff space (the analytification of K n), which contains themetric space K n as a dense subspace whenever the ground field K is algebraicallyclosed. His construction consists in replacing each point x of a given K -variety withthe space of all rank one valuations on the residue field κ(x) that extend v. Further, thetheory of stably dominated types, developed by Hrushovski–Loeser [23], deals withnon-archimedean fields with valuation of arbitrary rank and generalizes that of tametopology for Berkovich spaces. Currently, various analytic structures over Henselianrank one valued fields are intensively investigated (see, e.g., [11,12] for more infor-mation and [34] for the case of algebraically closed valued fields).

2 Quantifier elimination and cell decomposition

We begin with quantifier elimination due to Pas in the language L of Denef–Pas withthree sorts: the valued field K -sort, the value group �-sort and the residue field k-sort. The language of the K -sort is the language of rings; that of the �-sort is anyaugmentation of the language of ordered abelian groups (and ∞); finally, that of thek-sort is any augmentation of the language of rings. We denote K -sort variables byx, y, z, . . ., k-sort variables by ξ, ζ, η, . . ., and �-sort variables by k, q, r, . . ..

In the case of non-algebraically closed fields, passing to the three sorts with addi-tional two maps: the valuation v and the residue map, is not sufficient. Quantifierelimination due to Pas holds for Henselian valued fields of equicharacteristic zero inthe above three-sorted language with additional two maps: the valuation map v fromthe field sort to the value group and a map ac from the field sort to the residue field(angular component map) which is multiplicative, sends 0 to 0 and coincides with theresidue map on units of the valuation ring R of K .

Not all valued fields K have an angular component map, but it exists if K has across section, which happens whenever K is ℵ1-saturated (cf. [9, Chap. II]). More-over, a valued field K has an angular component map whenever its residue field k


is ℵ1-saturated (cf. [42, Corollary 1.6]). In general, unlike for p-adic fields and theirfinite extensions, adding an angular component map does strengthen the family ofdefinable sets. For both p-adic fields (Denef [13]) and Henselian equicharacteristiczero valued fields (Pas [41]), quantifier elimination was established by means of celldecomposition and a certain preparation theorem (for polynomials in one variable withdefinable coefficients) combined with each other. In the latter case, however, cells areno longer finite in number, but parametrized by residue field variables. In the proof ofthe closedness theorem, which is a fundamental tool for many results of this paper,we may use an angular component map because a given valued field can always bereplaced with an ℵ1-saturated elementary extension.

Finally, let us mention that quantifier elimination based on the sort RV := K ∗/(1+m)∪{0} (where K ∗ := K\{0} andm is the maximal ideal of the valuation ring R) wasintroduced by Besarab [4]. This new sort binds together the value group and residuefield into one structure. In the paper [22, Sect. 12], quantifier elimination for Henselianvalued fields of equicharacteristic zero, based on this sort, was derived directly fromthat by Robinson [43] for algebraically closed valued fields. Yet another, more generalresult, including Henselian valued fields of mixed characteristic, was achieved byCluckers–Loeser [10] for so-called b-minimal structures (from “ball minimal”); in thecase of valued fields, however, countably many sorts RVn := K ∗/(1 + nm) ∪ {0},n ∈ N, are needed.

Below we state the theorem on quantifier elimination due to Pas [41, Theorem 4.1].

Theorem 2.1 Let (K , �,k) be a structure for the three-sorted language L of Denef–Pas. Assume that the valued field K is Henselian and of equicharacteristic zero. Then(K , �,k) admits elimination of K -quantifiers in the language L.

We immediately obtain the following

Corollary 2.2 The three-sorted structure (K , �,k) admits full elimination of quanti-fiers whenever the theories of the value group � and the residue field k admit quantifierelimination in the languages of their sorts.

Below we prove another consequence of elimination of K -quantifiers, which willbe applied to the study of definable functions of one variable in Sect. 5.

Corollary 2.3 Let f : A → K be an L-definable function on a subset A of K n. Thenthere is a finite partition of A into L-definable sets Ai and irreducible polynomialsPi (x, y), i = 1, . . . , k, such that for each a ∈ Ai the polynomial Pi (a, y) in y doesnot vanish and

Pi (a, f (a)) = 0 for all a ∈ Ai , i = 1, . . . , k.

Proof By elimination of K -quantifiers, the graph of f is a finite union of sets Bi ,i = 1, . . . , k, defined by conditions of the form

(v( f1(x, y)), . . . , v( fr (x, y))) ∈ P, (ac g1(x, y), . . . , ac gs(x, y)) ∈ Q,

460 K. J. Nowak

where fi , g j ∈ K [x, y] are polynomials, and P and Q are L-definable subsets of�r and k

s , respectively. Each set Bi is the graph of the restriction of f to an L-definable subset Ai . Since, for each point a ∈ Ai , the fiber of Bi over a consists ofone point, the above condition imposed on angular components includes one of theform ac g j (x, y) = 0 or, equivalently, g j (x, y) = 0, for some j = 1, . . . , s, whichmay depend on a, where the polynomial g j (a, y) in y does not vanish. This meansthat the set

{(ac g1(x, y), . . . , ac gs(x, y)) : (x, y) ∈ Bi }

is contained in the union of hyperplanes⋃s

j=1{ξ j = 0} and, furthermore, that for eachpoint a ∈ Ai there is an index j = 1, . . . , s such that the polynomial g j (a, y) in ydoes not vanish and g j (a, f (a)) = 0. Clearly, for any j = 1, . . . , s, this property ofpoints a ∈ Ai is L-definable. Therefore, we can partition the set Ai into subsets eachof which fulfills the condition required in the conclusion with some irreducible factorsof the polynomial g j (x, y). �

Recall now some notation concerning cell decomposition. Consider anL-definablesubset D of K n × k

m , three L-definable functions

a(x, ξ), b(x, ξ), c(x, ξ) : D → K

and a positive integer ν. For each ξ ∈ km set

C(ξ) := {(x, y) ∈ K nx × Ky : (x, ξ) ∈ D,

v(a(x, ξ)) �1 v((y − c(x, ξ))ν) �2 v(b(x, ξ)), ac(y − c(x, ξ)) = ξ1},

where �1,�2 stand for <,≤ or no condition in any occurrence. If the sets C(ξ),ξ ∈ k

m , are pairwise disjoint, the union

C :=⋃

ξ∈km

C(ξ)

is called a cell in K n × K with parameters ξ and center c(x, ξ); C(ξ) is called a fiberof the cell C .

Theorem 2.4 (Preparation Cell Decomposition, [41, Theorem 3.2]) Let

f1(x, y), . . . , fr (x, y)

be polynomials in one variable y with coefficients being L-definable functions onK n

x . Then K n × K admits a finite partition into cells such that on each cell C withparameters ξ and center c(x, ξ) and for all i = 1, . . . , r we have:

v( fi (x, y)) = v(

fi (x, ξ)(y − c(x, ξ))νi)

, ac fi (x, y) = ξμ(i),


where fi (x, ξ) are L-definable functions, νi ∈ N for all i = 1, . . . , r , and the mapμ : {1, . . . , r} → {1, . . . , m} does not depend on x, y, ξ .

Remark 2.5 The functions f1(x, y), . . . , fr (x, y) are said to be prepared with respectto the variable y.

Every divisible ordered group � admits quantifier elimination in the language (<

,+,−, 0) of ordered groups. Therefore, it is not difficult to deduce from Theorems 2.1and 2.4 the following

Corollary 2.6 (Cell decomposition) If, in addition, the value group � is divisible,then every L-definable subset B of K n × K is a finite disjoint union of cells.

Every archimedean ordered group � (which of course may be regarded as a sub-group of the additive group R of real numbers) admits quantifier elimination in thePresburger language (<,+,−, 0, 1)with binary relation symbols≡n for congruencesmodulo n > 1, n ∈ N, where 1 denotes the minimal positive element of � if it existsor 1 = 0 otherwise. Under the circumstances, one can deduce in a similar manner thefollowing

Corollary 2.7 If, in addition, the valuation v is of rank 1, then every L-definablesubset B of K n × K is a finite disjoint union of sets each of which is a subset

F :=⋃

ξ∈km

F(ξ)

of a cell

C :=⋃

ξ∈km

C(ξ)

determined by finitely many congruences:

F(ξ) ={(x, y) ∈ C(ξ) : v

(fi (x, ξ)(y − c(x, ξ))ki

)≡M 0, i = 1, . . . , s

},

where fi are L-definable functions, ki ∈ N for i = 1, . . . , s, and M ∈ N, M > 1.

Remark 2.8 Corollary 2.7 will be applied to establish the closedness theorem (Theo-rem 3.1) in Sect. 7.

3 Closedness theorem

In this paper, we are interested mainly in geometry over a Henselian rank one valuedfield of equicharacteristic zero. Fromnowon,we shall assume (unless otherwise stated)that the ground field K is such a field. Below we state one of the basic theorems, onwhich many other results of our paper rely.

462 K. J. Nowak

Theorem 3.1 (Closedness theorem) Let D be an L-definable subset of K n. Then thecanonical projection

π : D × Rm −→ D

is definably closed in the K -topology, i.e., if B ⊂ D × Rm is an L-definable closedsubset, so is its image π(B) ⊂ D.

Observe that the K -topology is L-definable whence the above theorem is a first-order property. Therefore, it can be proven using elementary extensions, and thus, onemay assume that an angular component map exists. We shall provide two differentproofs for this theorem. The first, given in Sect. 4, is valid whenever the value group� is discrete, and is based on a procedure of algorithmic character. The other, given inSect. 7, is valid for the general case and makes use of Corollary 2.7 and fiber shrinkingfrom Sect. 6 which, in turn, relies on some results on L-definable functions of onevariable from Sect. 5. When the ground field K is locally compact, the closedness the-orem holds by a routine topological argument. We immediately obtain five corollariesstated below.

Corollary 3.2 Let D be an L-definable subset of K n and Pm(K ) stand for the pro-

jective space of dimension m over K . Then the canonical projection

π : D × Pm(K ) −→ D

is definably closed.

Corollary 3.3 Let A be a closed L-definable subset of Pm(K ) or Rm. Then every

continuous L-definable map f : A → K n is definably closed in the K -topology.

Corollary 3.4 Let φi , i = 0, . . . , m, be regular functions on K n, D be an L-definablesubset of K n and σ : Y −→ KA

n the blowup of the affine space KAn with respect to

the ideal (φ0, . . . , φm). Then the restriction

σ : Y (K ) ∩ σ−1(D) −→ D

is a definably closed quotient map.

Proof Indeed, Y (K ) can be regarded as a closed algebraic subvariety of K n ×Pm(K )

and σ as the canonical projection. �Since the problem is local with respect to the target space, the above corollary

immediately generalizes to the case where the K -variety Y is the blowup of a smoothK -variety X .

Corollary 3.5 Let X be a smooth K -variety, φi , i = 0, . . . , m, regular functions onX, D be an L-definable subset of X (K ) and σ : Y −→ X the blowup of the ideal(φ0, . . . , φm). Then the restriction

σ : Y (K ) ∩ σ−1(D) −→ D


is a definably closed quotient map.

Corollary 3.6 (Descent property)Under the assumptions of the above corollary, everycontinuous L-definable function

g : Y (K ) ∩ σ−1(D) −→ K

that is constant on the fibers of the blowup σ descends to a (unique) continuousL-definable function f : D −→ K .

4 Proof of Theorem 3.1 when the valuation is discrete

The proof given in this section is of algorithmic character. Through the transfer prin-ciple of Ax–Kochen–Ershov (see, e.g., [9]), it suffices to prove Theorem 3.1 for thecase where the ground field K is a complete, discretely valued field of equicharacter-istic zero. Such fields are, by virtue of Cohen’s structure theorem, the quotient fieldsK = k((t)) of formal power series rings k[[t]] in one variable t with coefficientsfrom a field k of characteristic zero. The valuation v and the angular component acof a formal power series are the degree and the coefficient of its initial monomial,respectively.

The additive group Z is an example of ordered Z -group, i.e., an ordered abeliangroupwith a (unique) smallest positive element (denoted by 1) subject to the followingadditional axioms:

∀ k k > 0 ⇒ k ≥ 1

and

∀ k ∃ qn−1∨

r=0

k = nq + r

for all integers n > 1. The language of the value group sort will be the Presburgerlanguage of ordered Z -groups, i.e., the language of ordered groups {<,+,−, 0} aug-mented by 1 and binary relation symbols ≡n for congruence modulo n subject to theaxioms:

∀ k, r k ≡n r ⇔ ∃ q k − r = nq

for all integers n > 1. This theory of ordered Z -groups has quantifier eliminationand definable Skolem (choice) functions. We can replace the above two countableaxiom schemas with universal ones after adding the unary function symbols

[ kn

]of

one variable k for division by n with remainder, which fulfill the following postulates:

[k

n

]

= q ⇔n−1∨

r=0

k = nq + r

464 K. J. Nowak

for all integers n > 1. The theory of ordered Z -groups admits therefore both quantifierelimination and universal axioms in the Presburger language augmented by divisionwith remainder. Thus every definable function is piecewise given by finitely manyterms and, consequently, is piecewise linear.

In the residue field sort, we can add new relation symbols for all definable sets andimpose suitable postulates. This enables quantifier elimination for the residue fieldin the augmented language. In this fashion, we have full quantifier elimination in thethree-sorted structure (K ,Z,k) with K = k((t)).

Now we can readily pass to the proof of Theorem 3.1 which, of course, reduceseasily to the case m = 1. So let B be anL-definable closed (in the K -topology) subsetof D × Ry ⊂ K n

x × Ry . It suffices to prove that if a lies in the closure of the projectionA := π(B), then there is a point b ∈ B such that π(b) = a.

Without loss of generality, we may assume that a = 0. Put

� := {(v(x1), . . . , v(xn)) ∈ Zn : x = (x1, . . . , xn) ∈ A}.

The set � contains points all coordinates of which are arbitrarily large, because thepoint a = 0 lies in the closure of A. Hence and by definable choice, � contains a set�0 of the form

�0 = {(k, α2(k), . . . , αn(k)) ∈ Nn : k ∈ �} ⊂ �,

where � ⊂ N is an unbounded definable subset and

α2, . . . , αn : � −→ N

are increasing unbounded functions given by a term (because a function in one variablegivenby a term is either increasingor decreasing).Weare going to recursively constructa point b = (0, w) ∈ B with w ∈ R by performing the following procedure ofalgorithmic character.

Step 1 Let

�1 := {(v(x1), . . . , v(xn), v(y)) ∈ �0 × N : (x, y) ∈ B},

and

β1(k) := sup {l ∈ N : (k, α2(k), . . . , αn(k), l) ∈ �1} ∈ N ∪ {∞}, k ∈ �0.

If lim supk→∞ β1(k) = ∞, there is a sequence (x (ν), y(ν)) ∈ B, ν ∈ N, such that

v(x (ν)1 ), . . . , v(x (ν)

n ), v(y(ν)) → ∞

when ν → ∞. Since the set B is a closed subset of D × Ry , we get

(x (ν), y(ν)) → 0 ∈ B when ν → ∞,


and thus w = 0 is the point we are looking for. Here the process stops. Otherwise

�1 × {l1} ⊂ �1

for some infinite definable subset �1 of �0 and l1 ∈ N. The set

{(v(x1), . . . , v(xn); ac(y)) ∈ �1 × k : (x, y) ∈ B, v(y) = l1}

is definable in the languageL. By full quantifier elimination, it is given by a quantifier-free formula with variables only from the value group �-sort and the residue fieldk-sort. Therefore, there is a finite partitioning of �1 into definable subsets over eachof which the fibers of the above set are constant, because quantifier-free L-definablesubsets of the productZn ×k of the two sorts are finite unions of the Cartesian productsof definable subsets in Z

n and in k, respectively. One of those definable subsets, say�′

1, must be infinite. Consequently, for some ξ1 ∈ k, the set

�2 := {(v(x1), . . . , v(xn), v(y − ξ1t l1)) ∈ �′1 × N : (x, y) ∈ B}

contains points of the form (k, l) ∈ Nn+1, where k ∈ �′

1 and l > l1.Step 2 Let

β2(k) := sup {l ∈ N : (k, α2(k), . . . , αn(k), l) ∈ �2} ∈ N ∪ {∞}, k ∈ �′1.

If lim supk→∞ β2(k) = ∞, there is a sequence (x (ν), y(ν)) ∈ B, ν ∈ N, such that

v(

x (ν)1

), . . . , v

(x (ν)

n

), v(

y(ν) − ξ1t l1)

→ ∞

when ν → ∞. Since the set B is a closed subset of D × Ry , we get

(x (ν), y(ν)

)→

(0, ξ1t l1

)∈ B when ν → ∞,

and thus w = ξ1t l1 is the point we are looking for. Here the process stops. Otherwise

�2 × {l2} ⊂ �2

for some infinite definable subset �2 of �′1 and l2 > l1. Again, for some ξ2 ∈ k, the

set

�3 := {(v(x1), . . . , v(xn), v(

y − ξ1t l1 − ξ2t l2))

∈ �′2 × N : (x, y) ∈ B}

contains points of the form (k, l) ∈ Nn+1, where k ∈ �′

2, �′2 is an infinite definable

subset of �2 and l > l2.Step 3 is carried out in the same way as the previous ones; and so on.

466 K. J. Nowak

In this fashion, the process either stops after a finite number of steps and then yieldsthe desired point w ∈ R (actually, w ∈ k[t]) such that (0, w) ∈ B, or it does not stopand then yields a formal power series

w := ξ1t l1 + ξ2t l2 + ξ3t l3 + . . . , 0 ≤ l1 < l2 < l3 < . . .

such that for each ν ∈ N there exists an element (x (ν), y(ν)) ∈ B for which

v(y(ν) − ξ1t l1 − ξ2t l2 − . . . − ξν t lν ) ≥ lν + 1 ≥ ν, v(x (ν)1 ), . . . , v(x (ν)

1 ) ≥ ν.

Hencev(y(ν)−w) ≥ ν, and thus the sequence (x (ν), y(ν)) tends to the pointb := (0, w)

when ν tends to ∞. Since the set B is a closed subset of D × R, the point b belongsto B, which completes the proof.

5 Definable functions of one variable

Consider first a complete rank one valued field L . For every nonnegative integer r , letL{x}r be the local ring of all formal power series

φ(x) =∞∑

k=0

ak xk ∈ L[[x]]

in one variable x such that v(ak) + kr tends to ∞ when k → ∞; L{x}0 coincideswith the ring of restricted formal power series. Then the local ring

L{x} :=∞⋃

r=0

L{x}r

is Henselian, which can be directly deduced bymeans of the implicit function theoremfor restricted power series in several variables (see [7, Chap. III, Sect. 4], [17] andalso [19, Chap. I, Sect. 5]).

We keep the assumption that the ground field K is a Henselian rank one valued fieldof equicharacteristic zero. Let L be the completion of the algebraic closure K of K .Clearly, the Henselian local ring L{x} is closed under division by the coordinate andpower substitution. Therefore, it follows from our paper [38, Sect. 2] that Puiseux’stheorem holds for L{x}. We still need an auxiliary lemma.

Lemma 5.1 The field K is a closed subspace of its algebraic closure K .

Proof This follows directly from that the field K is algebraically maximal (as it isHenselian and finitely ramified; see, e.g., [15, Chap. 4]), but can also be shown asfollows.Denote by cl (E, F) the closure of a subset E in F , and let K be the completionof K . We have

cl (K , K ) = cl (K , L) ∩ K = K ∩ K .


Now, through the transfer principle of Ax–Kochen–Ershov (see, e.g., [9]), K is anelementary substructure of K and, a fortiori, is algebraically closed in K . Hencecl (K , K ) = K ∩ K = K , as asserted.

Now consider an irreducible polynomial

P(x, y) =d∑

i=0

pi (x)yi ∈ K [x, y]

in two variables x, y of y-degree d ≥ 1. Let Z be the Zariski closure of its zero locusin K × P

1(K ). Performing a linear fractional transformation over the ground field Kof the variable y, we can assume that the fiber {w1, . . . , ws}, s ≤ d, of Z over x = 0does not contain the point at infinity, i.e.,w1, . . . , ws ∈ K . Then pd(0) �= 0 and pd(x)

is a unit in L{x}. Via Hensel’s lemma, we get the Hensel decomposition

P(x, y) = pd(x) ·s∏

j=1

Pj (x, y)

of P(x, y) into polynomials

Pj (x, y) = (y − w j )d j + p j1(x)(y − w j )

d j−1 + · · · + p jd j (x) ∈ L{x}[y − w j ]

which areWeierstrass with respect to y −w j , j = 1, . . . , s, respectively. By Puiseux’stheorem, there is a neighborhood U of 0 ∈ K such that the trace of Z on U × K is afinite union of sets of the form

Zφ j = {(xq , φ j (x)) : x ∈ U } with φ j ∈ L{x}, q ∈ N, j = 1, . . . , s.

Obviously, for j = 1, . . . , s, the fiber of Zφ j over x ∈ U tends to the pointφ j (0) = w j

when x → 0.If φ j (0) ∈ K\K , it follows from Lemma 5.1 that

Zφ j ∩ ((U ∩ K ) × K ) = ∅,

after perhaps shrinking the neighborhood U .Let us mention that if

φ j (0) ∈ K and φ j ∈ L{x}\K {x},

then

Zφ j ∩ ((U ∩ K ) × K ) = {(0, φ j (0))}

468 K. J. Nowak

after perhaps shrinking the neighborhood U . Indeed, let

φ j (x) =∞∑

k=0

ak xk ∈ L[[x]]

and p be the smallest positive integer with ap ∈ L\K . Since K is a closed subspaceof L , we get

∞∑

k=p

ak xk = x p

⎛

⎝ap + x ·∞∑

k=p+1

ak xk−(p+1)

⎞

⎠ /∈ K

for x close enough to 0, and thus the assertion follows.Suppose now that an L-definable function f : A → K satisfies the equation

P(x, f (x)) = 0 for x ∈ A

and 0 is an accumulation point of the set A. It follows immediately from the foregoingdiscussion that the set A can be partitioned into a finite number of L-definable setsA j , j = 1, . . . , r with r ≤ s, such that, after perhaps renumbering of the fiber{w1, . . . , ws} of the set {P(x, f (x)) = 0} over x = 0, we have

limx→0

f |A j (x) = w j for each j = 1, . . . , r.

Hence and by Corollary 2.3, we immediately obtain the following

Proposition 5.2 (Existence of the limit) Let f : A → K be an L-definable functionon a subset A of K and suppose 0 is an accumulation point of A. Then there is a finitepartition of A into L-definable sets A1, . . . , Ar and points w1 . . . , wr ∈ P

1(K ) suchthat

limx→0

f |A j (x) = w j for j = 1, . . . , r.

Moreover, there is a neighborhood U of 0 such that each definable set

{(v(x), v( f (x))) : x ∈ (A j ∩ U )\{0}} ⊂ � × (� ∪ {∞}), j = 1, . . . , r,

is contained in an affine line with rational slope

l = p j

q· k + β j , j = 1, . . . , r,

with p j , q ∈ Z, q > 0, β j ∈ �, or in � × {∞}.


Remark 5.3 Note that the first conclusion (existence of the limit) could also be estab-lished via the lemma on the continuity of roots of a monic polynomial (which canbe found in, e.g., [6, Chap. 3, Sect. 4]). Yet another approach for the case of tametheories is provided in [18, Lemma 2.20]. The second conclusion relies on Puiseux’sparametrization.

6 Fiber shrinking for definable sets

Let A be an L-definable subset of K n with accumulation point

a = (a1, . . . , an) ∈ K n

and E an L-definable subset of K with accumulation point a1. We call an L-definablefamily of sets

� =⋃

t∈E

{t} × �t ⊂ A

an L-definable x1-fiber shrinking for the set A at a if

limt→a1

�t = (a2, . . . , an),

i.e., for any neighborhood U of (a2, . . . , an) ∈ K n−1, there is a neighborhood V ofa1 ∈ K such that ∅ �= �t ⊂ U for every t ∈ V ∩ E , t �= a1. When n = 1, A is itselfa fiber shrinking for the subset A of K at an accumulation point a ∈ K . This conceptis a relaxed version of curve selection. It is used in Sects. 7 and 8 in the proofs of theclosedness theorem and a certain version of curve selection.

Proposition 6.1 (Fiber shrinking) Every L-definable subset A of K n with accumula-tion point a ∈ K n has, after a permutation of the coordinates, an L-definable x1-fibershrinking at a.

Proof We proceed with induction with respect to the dimension of the ambient affinespace n. The case n = 1 is trivial. So assuming the assertion to hold for n, we shallprove it for n + 1. We may, of course, assume that a = 0. Let x = (x1, . . . , xn+1) becoordinates in K n

x .If 0 is an accumulation point of the intersections

A ∩ {xi = 0}, i = 1, . . . , n + 1,

we are done by the induction hypothesis. Thus we may assume that the intersection

A ∩n+1⋃

i=1

{xi = 0} = ∅

470 K. J. Nowak

is empty. Then the definable (in the �-sort) set

P := {(v(x1), . . . , v(xn+1)) ∈ �n+1 : x ∈ A}

has an accumulation point (∞, . . . ,∞).Since the �-sort admits quantifier elimination in the language of ordered groups

augmented by binary relation symbols ≡n for congruence modulo n, every definablesubset of �n+1 is a finite union of subsets of semilinear sets contained in �n+1 thatare determined by a finite number of congruences

n+1∑

j=1

ri j · k j ≡N αi , i = 1, . . . , s; (6.1)

here N ∈ N, N > 1, ri j ∈ Z, αi ∈ � for i = 1, . . . , s, j = 1, . . . , n + 1.Consequently, there exists a semilinear subset P0 of Rn+1 given by finitely many

linear equations and inequalities with integer coefficients andwith constant terms from� such that the subset P1 of P0 ∩ �n+1 determined by congruences of the form 6.1is contained in P and has an accumulation point (∞, . . . ,∞). Therefore, there existsan affine semiline

L := {(r1 · k + γ1, . . . , rn+1 · k + γn+1) : k ∈ �, k ≥ 0} ,

where r1, . . . , rn+1 are positive integers, passing through a point

(γ1, . . . , γn+1) ∈ P1 ⊂ �n+1

and contained in P0. It is easy to check that the set

L0 := {(γ1 + rr1N , . . . , γn+1 + rrn+1N ) : r ∈ N} ⊂ P1

is contained in P1. Then

� := {x ∈ A : (v(x1), . . . , v(xn+1)) ∈ L0}

is an L-definable x1-fiber shrinking for the set A at 0. This finishes the proof. �

7 Proof of Theorem 3.1 for the general case

The proof reduces easily to the case m = 1. We must show that if B is an L-definablesubset of D × R and a point a lies in the closure of A := π(B), then there is a pointb in the closure of B such that π(b) = a. We may obviously assume that a = 0 /∈ A.By Proposition 6.1, there exists, after a permutation of the coordinates, anL-definable


x1-fiber shrinking � for A at a:

� =⋃

t∈E

{t} × �t ⊂ A, limt→0

�t = 0;

here E is the canonical projection of A onto the x1-axis. Put

B∗ := {(t, y) ∈ K × R : ∃ u ∈ �t (t, u, y) ∈ B};

it easy to check that if a point (0, w) ∈ K 2 lies in the closure of B∗, then the point(0, w) ∈ K n+1 lies in the closure of B. The problem is thus reduced to the case n = 1and a = 0 ∈ K .

By Corollary 2.7, we can assume that B is a subset F of a cell C

F ⊂ C ⊂ Kx × R ⊂ Kx × Ky

of the form

F(ξ) := {(x, y) ∈ Kx × Ky : (x, ξ) ∈ D,

v(a(x, ξ)) �1 v((y − c(x, ξ))ν) �2 v(b(x, ξ)), ac(y − c(x, ξ)) = ξ1,

v(

fi (x, ξ)(y − c(x, ξ))ki)

≡M 0, i = 1, . . . , s}.

But the set

{(v(x), ξ) ∈ � × km : ∃ y ∈ R (x, y) ∈ F(ξ)}

is anL-definable subset of the product�×km of the two sorts, which is, by elimination

of K -quantifiers, a finite union of the Cartesian products of definable subsets in � andinkm , respectively. It follows that 0 is an accumulation point of the projectionπ(F(ξ ′))of the fiber F(ξ ′) for a parameter ξ ′ ∈ k

m . We are thus reduced to the case whereB is the fiber F(ξ ′) of the set F for a parameter ξ ′. For simplicity, we abbreviatec(x, ξ ′), a(x, ξ ′), b(x, ξ ′) and fi (x, ξ ′) to c(x), a(x), b(x) and fi (x), i = 1, . . . , s.Denote by E ⊂ K the domain of these functions; then 0 is an accumulation point ofE .

In the statement of Theorem 3.1, we may equivalently replace R with the projectiveline P1(K ), because the latter is the union of two open and closed charts biregular toR. By Proposition 5.2, we can thus assume that the limits, say c(0), a(0), b(0), fi (0)of c(x), a(x), b(x), fi (x) (i = 1, . . . , s) when x → 0 exist in P1(K ) and, moreover,there is a neighborhood U of 0 such that, each definable set

{(v(x), v( fi (x))) : x ∈ (E ∩ U )\{0}} ⊂ � × (� ∪ {∞}), i = 1, . . . , s,

is contained in an affine line with rational slope

l = pi

q· k + βi , i = 1, . . . , s, (7.1)

472 K. J. Nowak

with pi , q ∈ Z, q > 0, βi ∈ �, or in � × {∞}.Performing a linear fractional transformation of the coordinate y, we get

c(0), a(0), b(0) ∈ K .

The role of the center c(x) is immaterial. We can assume, without loss of generality,that it vanishes, c(x) ≡ 0, for if a point b = (0, w) ∈ K 2 lies in the closure of the cellwith zero center, the point (0, w + c(0)) lies in the closure of the cell with center c(x).

When �1 occurs and a(0) = 0, the set F(ξ ′) is itself an x-fiber shrinking at (0, 0)and the point b = (0, 0) is an accumulation point of B lying over a = 0, as desired.

So suppose that either only �2 occurs or �1 occurs and a(0) �= 0. By eliminationof K -quantifiers, the set v(E) is a definable subset of �. The value group � admitsquantifier elimination in the language of ordered groups augmented by symbols ≡n

for congruences modulo n, n ∈ N, n > 1 (cf. Sect. 2). Therefore, the set v(E) is ofthe form

v(E) = {k ∈ (α,∞) ∩ � : m j k ≡N γ j , j = 1, . . . , t}, (7.2)

where α, γ j ∈ �, m j ∈ N for j = 1, . . . , t .Now, take an element (u, w) ∈ F(ξ ′) with u ∈ (E ∩U )\{0}. By equality 7.2, there

is a point xr ∈ E , r ∈ N, with

v(ur ) = v(u) + rq M N .

By equality 7.1, we get

v( fi (ur )) = v( fi (u)) + r pi M N , i = 1, . . . , s.

Hence

v(

fi (ur )wki)

= v( fi (ur )) + kiv(w)

= v( fi (u)) + r pi M N + kiv(w)

= v(

fi (u)wki)

+ r pi M N ≡M 0. (7.3)

Of course, after shrinking the neighborhood U , we may assume that v(a(x)) =v(a(0)) < ∞ for all x ∈ (E ∩ U )\{0}. Consequently,

v(a(ur )) �1 v(wν) �2 v(b(ur )).

Hence and by 7.3, we get (ur , w) ∈ F(ξ ′). Since ur tends to 0 ∈ K when r → ∞,the point (0, w) is an accumulation point of F(ξ ′) lying over 0 ∈ K , which completesthe proof of the closedness theorem.


8 Curve selection

We now pass to curve selection over non-locally compact ground fields under study.While the real version of curve selection goes back to the papers [8,45] (see also [5,35,36]), the p-adic one was achieved in the papers [14,44]. Before proving a generalversion for L-definable sets, we give a version for valuative semialgebraic sets. Ourapproach relies on resolution of singularities, which was already suggested by Denef–van den Dries [14] in the remark after Theorem 3.34.

By a valuative semialgebraic subset of K n , wemean a (finite) Boolean combinationof elementary valuative semialgebraic subsets, i.e., sets of the form

{x ∈ K n : v( f (x)) ≤ v(g(x))

},

where f and g are regular functions on K n . We call a map ϕ semialgebraic if its graphis a valuative semialgebraic set.

Proposition 8.1 Let A be a valuative semialgebraic subset of K n. If a point a ∈ K n

lies in the closure (in the K -topology) of A\{a}, then there is a semialgebraic mapϕ : R −→ K n given by restricted power series such that

ϕ(0) = a and ϕ(R\{0}) ⊂ A\{a}.

Proof It is easy to check that every valuative semialgebraic set is a finite union ofbasic valuative semialgebraic sets, i.e., sets of the form

{x ∈ K n : v( f1(x)) �1 v(g1(x)), . . . , v( fr (x)) �r v(gr (x))},

where f1, . . . , fr , g1, . . . , gr are regular functions and �1, . . . ,�r stand for ≤ or<. We may assume, of course, that A is a set of this form and a = 0. Take a finitecomposite

σ : Y −→ KAn

of blowups along smooth centers such that the pullbacks of the coordinates x1, . . . , xn

and the pullbacks

f σ1 := f1 ◦ σ, . . . , f σ

r := fr ◦ σ and gσ1 := g1 ◦ σ, . . . , gσ

r := gr ◦ σ

are normal crossing divisors ordered with respect to divisibility relation, unless theyvanish. Since the restriction σ : Y (K ) −→ K n is definably closed (Corollary 3.5),there is a point b ∈ Y (K ) ∩ σ−1(a) which lies in the closure of the set

B := Y (K ) ∩ σ−1(A\{a}).

Further, we get

Y (K ) ∩ σ−1(A) = {v( f σ

1 (y)) �1 v(gσ1 (y))

} ∩ . . . ∩ {v( f σr (y)) �r v

(gσ

r (y))}

,

474 K. J. Nowak

and thus σ−1(A) is in suitable local coordinates y = (y1, . . . , yn) near b = 0 a finiteintersection of sets of the form

{v(yα) ≤ v(u(y))}, {v(u(y)) ≤ v(yα)

},{v(yβ) < ∞}

or {∞ = v(yγ )},

where α, β, γ ∈ Nn and u(y) is a regular, nowhere vanishing function.

The first case cannot occur because b = 0 lies in the closure of B; the second caseholds in a neighborhood of b; the third and fourth cases are equivalent to yβ �= 0 andyγ = 0, respectively. Consequently, since the pullbacks of the coordinates x1, . . . , xn

are monomial divisors too, B contains the set (R\{0}) · c when c ∈ B is a pointsufficiently close to b = 0. Then the map

ϕ : R −→ K n, ϕ(z) = σ(z · c)

has the desired properties. �We now pass to the general version of curve selection for L-definable sets.

Proposition 8.2 Let A be an L-definable set subset of K n. If a point a ∈ K n liesin the closure (in the K -topology) of A\{a}, then there exist a semialgebraic mapϕ : R −→ K n given by restricted power series and an L-definable subset E of R withaccumulation point 0 such that

ϕ(0) = a and ϕ(E\{0}) ⊂ A\{a}.

Proof We proceed with induction with respect to the dimension of the ambient spacen. The case n = 1 being evident, suppose n > 1. By elimination of K -quantifiers,similarly as in Sect. 2, the set A\{a} is a finite union of sets defined by conditions ofthe form

(v( f1(x)), . . . , v( fr (x))) ∈ P, (ac g1(x), . . . , ac gs(x)) ∈ Q,

where fi , g j ∈ K [x] are polynomials, and P and Q are definable subsets of �r andk

s , respectively (since x = 0 iff ac x = 0). Thus we may assume that A is such a setand, of course, that a = 0.

Again, take a finite composite

σ : Y −→ KAn

of blowups along smooth centers such that the pullbacks

f σ1 , . . . , f σ

r and gσ1 , . . . , gσ

r

are normal crossing divisors unless they vanish. Since the restriction σ : Y (K ) −→K n is definably closed (Corollary 3.5), there is a point b ∈ Y (K )∩σ−1(a) which liesin the closure of the set

B := Y (K ) ∩ σ−1(A\{a}).


Take local coordinates y1. . . . , yn near b in which b = 0 and every pullback above isa normal crossing. We shall first select a semialgebraic map ψ : R −→ Y (K ) givenby restricted power series and an L-definable subset E of R with accumulation point0 such that

ψ(0) = b and ψ(E\{0}) ⊂ B.

Since the valuation map and the angular component map composed with a con-tinuous function are locally constant near any point at which this function does notvanish, the conditions which describe the set B near b are of the form

(v(y1), . . . , v(yn)) ∈ P, (ac y1, . . . , ac yn) ∈ Q,

where P and Q are definable subsets of �n and kn , respectively.

The set B0 determined by the conditions

(v(y1), . . . , v(yn)) ∈ P,

(ac y1, . . . , ac yn) ∈ Q ∩n⋃

i=1

{ξi = 0},

is contained near b in the union of hyperplanes {yi = 0}, i = 1, . . . , n. If b is anaccumulation point of the set B0, then the desired map ψ exists by the inductionhypothesis. Otherwise b is an accumulation point of the set B1 := B\B0.

Analysis from the proof of Proposition 6.1 (fiber shrinking) shows that the congru-ences describing the definable subset P of�n are not an essential obstacle to finding thedesired mapψ , but affect only the definable subset E of R. Neither are the conditions

Q \n⋃

i=1

{ξi = 0}

imposed on the angular components of the coordinates y1, . . . , yn , because then noneof them vanishes. Therefore, in order to select the map ψ , we must first of all analyzethe linear conditions (equalities and inequalities) describing the set P .

The set P has an accumulation point (∞, . . . ,∞) as b = 0 is an accumulationpoint of B. We see, similarly as in the proof of Proposition 6.1 (fiber shrinking), thatP contains a definable subset of a semiline

L := {(r1 · k + γ1, . . . , rn · k + γn) : k ∈ �, k ≥ 0} ,

where r1, . . . , rn are positive integers, passing through a point

γ1, . . . , γn ∈ P ⊂ �n;

clearly, (∞, . . . ,∞) is an accumulation point of that definable subset of L .

476 K. J. Nowak

Now, take some elements

(ξ1, . . . , ξn) ∈ Q \n⋃

i=1

{ξi = 0}

and next some elements w1, . . . , wn ∈ K for which

v(w1) = γ1, . . . , v(wn) = γn and ac w1 = ξ1, . . . , ac wn = ξn .

There exists an L-definable subset E of R which is determined by some congruencesimposed on v(t) (as in the proof of Proposition 6.1) and the conditions ac t = 1 suchthat the subset

F := {(w1 · tr1 , . . . , wn · trn

) : t ∈ E}

of the arc

ψ : R → Y, ψ(t) = (w1 · tr1, . . . , wn · trn

)

is contained in B1. Then ϕ := σ ◦ ψ is the map we are looking for. This completesthe proof. �

9 Łojasiewicz inequality

In this section, we provide certain general versions of the Łojasiewicz inequality. Forthe classical version over the real ground field, we refer the reader to [5, Thm. 2.6.6].

Proposition 9.1 Let f, g : A → K be two continuous L-definable functions on aclosed (in the K -topology) L-definable subset A of Rm. If

{x ∈ A : g(x) = 0} ⊂ {x ∈ A : f (x) = 0},

then there exist a positive integer s and a continuous L-definable function h on A suchthat f s(x) = h(x) · g(x) for all x ∈ A.

Proof It is easy to check that the set

Aγ := {x ∈ A : v( f (x)) = γ }

is a closed L-definable subset of A for every γ ∈ �. Hence and by Corollary 3.3 tothe closedness theorem, the set g(Aγ ) is a closedL-definable subset of K\{0}, γ ∈ �.The set v(g(Aγ )) is thus bounded from above, i.e.,

v(g(Aγ )) < α(γ )


for some α(γ ) ∈ �. By elimination of K -quantifiers, the set

� := {(v( f (x)), v(g(x))) ∈ �2 : x ∈ A} ⊂ {(γ, δ) ∈ �2 : δ < α(γ )}

is a definable subset of �2, and thus it is described by a finite number of linearinequalities and congruences. Hence

� ∩ {(γ, δ) ∈ �2 : γ > γ0} ⊂ {(γ, δ) ∈ �2 : δ < s · γ }

for a positive integer s and some γ0 ∈ �. We thus get

v(g(x)) < s · v( f (x)) if x ∈ A, v( f (x)) > γ0,

whence

v(g(x)) < v( f s(x)) if x ∈ A, v( f (x)) > γ0.

Consequently, the quotient f s/g extends by zero through the zero set of the denom-inator to a (unique) continuous L-definable function on A. This finishes the proof.

�The above theorem can be generalized as follows.

Proposition 9.2 Let U and F be two L-definable subsets of K m, suppose U is openand F closed in the K -topology and consider two continuous L-definable functionsf, g : A → K on the locally closed subset A := U ∩ F of K m. If

{x ∈ A : g(x) = 0} ⊂ {x ∈ A : f (x) = 0},

then there exist a positive integer s and a continuous L-definable function h on A suchthat f s(x) = h(x) · g(x) for all x ∈ A.

Proof We shall adapt the foregoing arguments. Since the setU is open, its complementV := K m\U is closed in K m and A is the following union of open and closed subsetsof K m and of Pm(K ):

Xβ := {x ∈ K m : v(x1), . . . , v(xm) ≥ −β,

v(x − y) ≤ β for all y ∈ V },

where β ∈ �, β ≥ 0. As before, we see that the sets

Aβ,γ := {x ∈ Xβ : v( f (x)) = γ } with β, γ ∈ �

are closed L-definable subsets of Pm(K ), and next that the sets g(Aβ,γ ) are closedL-definable subsets of K\{0} for all β, γ ∈ �. Likewise, we get

� := {(β, v( f (x)), v(g(x))) ∈ �3 : x ∈ Xβ} ⊂⊂ {(β, γ, δ) ∈ �3 : δ < α(β, γ )}

478 K. J. Nowak

for some α(β, γ ) ∈ �. Consequently, since � is a definable subset of �3, there exista positive integer s and elements γ0(β) ∈ � such that

� ∩ {(β, γ, δ) ∈ �3 : γ > γ0(β)} ⊂ {(β, γ, δ) ∈ �3 : δ < s · γ }.

Since A is the union of the sets Xβ , it is not difficult to check that the quotient f s/gextends by zero through the zero set of the denominator to a (unique) continuousL-definable function on A, which is the desired result. �

10 Extending continuous hereditarily rational functions

We first recall an elementary lemma from [27, Lemma 15].

Lemma 10.1 If the ground field K is not algebraically closed, then there are polyno-mials Gr (x1, . . . , xr ) in any number of variables whose only zero on K r is (0, . . . , 0).In particular,

G2(x1, x2) = xd1 + a1xd−1

1 x2 + · · · + ad xd2 ,

where

td + a1td−1 + · · · + ad ∈ K [t]

is a polynomial with no roots in K .

We keep further the assumption that K is a Henselian rank one valued field ofequicharacteristic zero and, additionally, that it is not algebraically closed. We haveat our disposal the descent property (Corollary 3.6) and the Łojasiewicz inequality(Proposition 9.2). Therefore, by adapting mutatis mutandis its proof, we are able tocarry over Proposition 11 from [27] on extending continuous hereditarily rationalfunctions (being its main extension result) to the case of such non-archimedean fields.We first recall the definition. Given a K -variety Z , we say that a continuous functionf : Z(K ) → K is hereditarily rational if every irreducible subvariety Y ⊂ Z has aZariski dense open subvariety Y 0 ⊂ Z such that f |Y 0(K ) is regular.

Theorem 10.2 Let X be a smooth K -variety and W ⊂ Z ⊂ X closed subvarieties.Let f be a continuous hereditarily rational function on Z(K ) that is regular at all K -points of Z(K )\W (K ). Then f extends to a continuous hereditarily rational functionF on X (K ) that is regular at all K -points of X (K )\W (K ).

Remark 10.3 The corresponding theorem for differentiable hereditarily rational func-tions remains an open problem as yet (cf. Remark 13.9 and the discussion afterward).

Sketch of the Proof We shall keep the notation from the paper [27]. The main mod-ification of the proof in comparison with that paper is the definition of the functions


G and F2n which improve the rational function P/Q. Now we need the followingcorrections:

G := P

Q· Qd

G2(Q, H)

and

Fdn := G · Qdn

G2(Qn, H)= P

Q· Qd

G2(Q, H)· Qdn

G2(Qn, H),

where the positive integer d and the polynomial G2 are taken from Lemma 10.1. Itis clear that the restriction of Fdn to Z\W equals f2, and thus Theorem 10.2 will beproven once we show that the rational function Fdn restricts to a continuous function�dn on X (K ) for n � 1. �

We work on the variety π : X1(K ) −→ X (K ) obtained by blowing up the ideal(P Qd−1, G2(Q, H)); let E := π−1(W ) be its exceptional divisor. Equivalently,X1(K ) is the Zariski closure of the graph of G in X (K ) × P

1(K ). Two open chartsare considered:

• a Zariski open neighborhood U∗ of the closure (in the K -topology) Z∗ ofπ−1(Z(K )\W (K ));

• an open (in the K -topology) set V ∗ := X1(K )\Z∗, which is anL-definable subsetof X1(K ).

Via the descent property (Corollary 3.6), it suffices to show that the rational functionFdn ◦ π (with n � 1) extends to a continuous function on X1(K ) that vanishes onE(K ).

The subtlest analysis is on the latter chart, on which Fdn ◦ π can be written in theform

Fdn ◦ π = (P ◦ π) ·(

Qdn−1

Hd◦ π

)

·(

Qd

G2(Q, H)◦ π

)

·(

Hd

G2(Qn, H)◦ π

)

.

Note that on V ∗ the function H ◦ π vanishes only along E(K ) and Q ◦ π vanishesalong E(K ) too. Therefore, we can apply Proposition 9.2 (Łojasiewicz inequality) tothe numerator and denominator of the first factor, which are regular functions on thechart V ∗, to immediately deduce that the first factor extends to a continuous rationalfunction on V ∗ that vanishes along E(K ) ∩ V ∗ for n � 1.

What still remains to prove (cf. [27]) is that the factors

Qdn

G2(Qn, H),

Qd

G2(Q, H)and

Hd

G2(Qn, H)

are regular functions off W (K ) whose valuations are bounded from below. But thisfollows immediately from an auxiliary lemma:

480 K. J. Nowak

Lemma 10.4 Let g be the polynomial from the proof of Lemma 10.1. Then the set ofvalues

v

(td

g(t)

)

∈ �, t ∈ K ,

is bounded from below.

In order to prove this lemma, observe that

v

(td

g(t)

)

= v(1 + a1

t+ · · · + ad

td

)

for t ∈ K , t �= 0. Hence the values under study are zero if iv(t) < v(ai ) for alli = 1, . . . , d. Therefore, we are reduced to analyzing the case where

v(t) ≥ k := min

{v(ai )

i: i = 1, . . . , d

}

.

Denote by� the value group of v. Thuswemust show that the set of values v(g(t)) ∈ �

when v(t) ≥ k is bounded from above.Take elements a, b ∈ R, a, b �= 0, such that aai ∈ R for all i = 1, . . . , d, and

bt ∈ R whenever v(t) ≥ k. Then

h(abt) := (ab)d g(t)

= (abt)d + aba1(abt)d−1 + (ab)2a2(abt)d−2 + · · · + (ab)dad

is a monic polynomial with coefficients from R which has no roots in K . Clearly, itis sufficient to show that the set of values v(h(t)) ∈ � when t ∈ R is bounded fromabove.

Consider a splitting field K = K (u1, . . . , ud) of the polynomial h, whereu1, . . . , ud are the roots of h. Let v be a (unique) extension to K of the valuationv, R be its valuation ring and � ⊃ � its value group (see, e.g., [46, Chap. VI, Sect. 11]for valuations of algebraic field extensions). Then

u1, . . . , ud ∈ R\R and h(t) =d∏

i=1

(t − ui ).

Since R is a closed subring of R by Lemma 5.1, there exists an l ∈ � such thatv(t − ui ) ≤ l for all i = 1, . . . , d and t ∈ R. Hence v(h(t)) ≤ dl for all t ∈ R, andthus the lemma follows.

In this fashion, we have demonstrated how to adapt the proof of Proposition 11from [27] to the case of Henselian rank one valued fields of equicharacteristic zero.Note that the proofs of all remaining results from that paper work over general topo-logical fields with a density property introduced in [27, Sect. 3] and recalled below.


A topological field K satisfies the density property if one of the following equivalentconditions holds:

(1) If X is a smooth, irreducible K -variety and ∅ �= U ⊂ X is a Zariski open subset,then U (K ) is dense in X (K ) in the K -topology.

(2) If C is a smooth, irreducible K -curve and ∅ �= C0 ⊂ C is a Zariski open subset,then C0(K ) is dense in C(K ) in the K -topology.

(3) If C is a smooth, irreducible K -curve, then C(K ) has no isolated points.

The examples of such fields are, in particular, all Henselian rank one valued fields.

11 Regulous functions and sets

In these last three sections, we shall carry the theory of regulous functions over thereal ground field R, developed by Fichou–Huisman–Mangolte–Monnier [16], overto non-archimedean algebraic geometry over Henselian rank one valued fields K ofequicharacteristic zero. We assume that the ground field K is not algebraically closed.(Otherwise, the notion of a regulous function on a normal variety coincides with thatof a regular function and, in general, the study of continuous rational functions leadsto the concept of seminormality and seminormalization; cf. [1,2] or [26, Sect. 10.2]for a recent treatment.) Every such field enjoys the density property. The K -pointsX (K ) of any algebraic K -variety X inherit from K a topology, called the K -topology.

In this section, we deal with the ground fields K with the density property. Observefirst that if f is a rational function on an affine K -variety X which is regular on aZariski open subset U , then there exist two regular functions p, q on X such that

f = p

qand q(x) �= 0 for all x ∈ U (K ).

When X ⊂ KAn , then p, q can be polynomial functions. For every rational function

f on X , there is a largest Zariski open subset of X on which f is regular, calledthe regular locus of f and denoted by dom ( f ). Further, assume that Z is a closedsubvariety of a K -variety X . Then every rational function f on Z that is regular onZ(K ) extends to a rational function F on X that is regular on X (K ). Both the resultscan be deduced via Lemma 10.1 (cf. [27], the proof of Lemma 15 on extending regularfunctions).

Suppose now that X is a smooth affine K -variety or, at least, an affine K -varietythat is smooth at all K -points X (K ). We say that a function f on X (K ) is k-regulous,k ∈ N ∪ {∞}, if it is of class Ck and there is a Zariski dense open subset U of X suchthat the restriction of f to U (K ) is a regular function. A function f on X (K ) is calledregulous if it is 0-regulous. Denote byRk(X (K )) the ring of k-regulous functions onX (K ). The ring R∞(X (K )) of ∞-regulous functions on X (K ) coincides with thering O(X (K )) of regular functions on X (K ). This follows easily from the faithfulflatness of the formal power series ring K [[x1, . . . , xn]] over the local ring of regularfunction germs at 0 ∈ K n . Therefore, we shall restrict ourselves to the case k ∈ N.

482 K. J. Nowak

When K is a Henselian rank one valued field of equicharacteristic zero, transforma-tion to a normal crossing by blowing up along smooth centers and the descent property(Corollary 3.6) enable the following characterization:

Given a smooth algebraic K -variety X , a function

f : X (K ) → K

is regulous iff there exists a finite composite σ : X → X of blowups with smoothcenters such that the pullback f σ := f ◦ σ is a regular function on X(K ).

We say that a subset V of K n is k-regulous closed if it is the zero set of a familyE ⊂ Rk(Kn) of k-regulous functions:

V = Z(E) := {x ∈ K n : f (x) = 0 for all f ∈ E}.

A subset U of K n is called k-regulous open if its complement Kn\U is k-regulousclosed. The family of k-regulous open subsets of K n is a topology on K n , called thek-regulous topology on K n .

If f �= 0 is a k-regulous function on K n with regular locus

U = dom ( f ) ⊂ K n,

then

f = p

qwhere p, q ∈ K [x1, . . . , xn], Z(q) = K n\U (K ),

and p, q are coprime polynomials. Clearly, Z(q) ⊂ Z(p) and it follows, by passageto the algebraic closure of K , that the zero set Z(q) is of codimension ≥ 2 in K n .Thus the complement K n\dom ( f ) is of codimension≥ 2 in K n . Consequently, everyk-regulous function on K is regular and every k-regulous function on K 2 is regular atall but finitely many points.

We now recall some results about algebraic varieties over arbitrary fields F . Let Vbe an affine F-variety. We are interested in the set V (F) of its F-points. Therefore,from now on, we shall (and may) assume that V (F) is Zariski dense in V . Then theregular locus Reg (V ) of V is a non-empty, Zariski open subset of V and, moreover, itstrace on the set V (F) is non-empty; cf. [32], Chap. VI, Corollary 1.17 to the Jacobiancriterion for regular local rings and the remark preceding it. If the ground field F isnot algebraically closed, then the trace Reg (V )∩ V (F) is smooth and affine, becauseit follows immediately from Lemma 10.1 that every algebraic subset of Fn is the zeroset of one polynomial. Summing up, we see that, for every affine F-subvariety V ofFA

n , the set V (F) ⊂ Fn of its F-points is a finite (disjoint) union of smooth, affine,Zariski locally closed subsets of Fn of pure dimension.

We call a subset E of K n constructible if it is a (finite) Boolean combination ofZariski closed subsets of K n . Every such set E is, of course, a finite union of Zariskilocally closed subsets. Further, in view of the foregoing discussion, E is a finite unionof smooth, affine, Zariski locally closed subsets of pure dimension with irreducible


Zariski closure. Thus it follows immediately from the density property that everyclosed (in the K -topology) constructible subset E of K n is a finite union of a uniqueirredundant family �(E) of constructible subsets each of which is the regular locusReg (V )∩K n of an irreducible affine K -subvarietyV of KA

n ; obviously,Reg (V )∩K n

is a smooth, Zariski locally closed subset of pure dimension dim V .Below we introduce the constructible topology on K n . We shall see in the next

section that the k-regulous topology coincides with the constructible topology for allk ∈ N.

Proposition 11.1 If K is a topological field with the density property, then the familyof all closed (in the K -topology) constructible subsets of K n is the family of closed setsfor a topology, called the constructible topology on K n. Furthermore, this topologyis noetherian, i.e., every descending sequence of closed constructible subsets of K n

stabilizes.

Proof Clearly, it suffices to prove only the last assertion.We shall follow the reasoningfrom our paper [37] which showed that the quasi-analytic topology is noetherian. Forany closed (in the K -topology) constructible subset E of K n , letμi (E) be the numberof elements from the family�(E), constructed above, of dimension i , i = 0, 1, . . . , n,and put

μ(E) = (μn(E), μn−1(E), . . . , μ0(E)) ∈ Nn+1.

Consider now a descending sequence of closed constructible subsets

K n ⊃ E1 ⊃ E2 ⊃ E3 ⊃ . . . .

It is easy to check that for any two closed (in the K -topology) constructible subsetsD ⊂ E , we have μ(D) ≤ μ(E) and, furthermore, D = E iff μ(D) = μ(E). Hencewe get the decreasing (in the lexicographic order) sequence of multi-indices

μ(E1) ≥ μ(E2) ≥ μ(E3) ≥ . . . ,

which must stabilize for some N ∈ N:

μ(EN ) = μ(EN+1) = μ(EN+2) = . . .

Then

EN = EN+1 = EN+2 = . . . ,

as desired. �Corollary 11.2 Suppose K is a field with the density property. Then there is a one-to-one correspondence between the irreducible closed constructible subsets E of K n

and the irreducible Zariski closed subsets V of K n:

α : E �−→ EZ

and β : V �−→ Reg (V )c,

484 K. J. Nowak

where EZ

stands for the Zariski closure of E and Ac

for the closure of A in theconstructible topology.

Proof We have α ◦ β = Id, because Reg (V )Z = V for every irreducible Zariski

closed subset V of K n . In view of the foregoing discussion, the assignment β issurjective. Therefore, every irreducible closed constructible subset E of K n is of theform E = β(V ) = Reg (V )

cfor an irreducible Zariski closed subset V of K n . Hence

(β ◦ α)(E) = (β ◦ α)(β(V )) = (β ◦ α ◦ β)(V ) = (β ◦ Id)(V ) = β(V ) = E,

and thus β ◦ α = Id, which finishes the proof. �Below we recall Proposition 8 from [27] which holds over any topological fields

with the density property.

Proposition 11.3 Let X be an algebraic K -variety and f a rational function onX that is regular on X0 ⊂ X. Assume that f |X0(K ) has a continuous extensionf c : X (K ) → K . Let Z ⊂ X be an irreducible subvariety that is not contained inthe singular locus of X. Then there is a Zariski dense open subset Z0 ⊂ Z such thatf c|Z0(K ) is a regular function.

We immediately obtain two corollaries:

Corollary 11.4 Let X be an algebraic K -variety that is smooth at all K -points X (K )

and f a rational function on X that is regular on X0 ⊂ X. Assume that f |X0(K )

has a continuous extension f c : X (K ) → K . Then there is a sequence of closedsubvarieties

∅ = X−1 ⊂ X0 ⊂ · · · ⊂ Xn = X

such that for i = 0, . . . , n the restriction of f to Xi (K )\Xi−1(K ) is regular. Moreover,we can require that each set Xi\Xi−1 be smooth of pure dimension i .

Corollary 11.5 If f is a regulous function on K n, then there is a sequence of Zariskiclosed subsets

∅ = E−1 ⊂ E0 ⊂ · · · ⊂ En = K n

such that for i = 0, . . . , n the restriction of f to Ei\Ei−1 is regular. Moreover, wecan require that each set Ei\Ei−1 be smooth of pure dimension i .

Remark 11.6 Given a finite number of regulous functions f1, . . . , f p, there is a filtra-tion

∅ = E−1 ⊂ E0 ⊂ · · · ⊂ En = K n

as in the above corollary such that for i = 0, . . . , n the restriction of each function f j ,j = 1, . . . , p, to Ei\Ei−1 is regular.


Now, three further consequences of the above corollary will be drawn. We say thata map

f = ( f1, . . . , f p) : K n → K p

is k-regulous if all its components f1, . . . , f p are k-regulous functions on K n .

Corollary 11.7 If two maps

g : K m → K n and f : K n → K p

are k-regulous, so is its composition f ◦ g.

Proof Indeed, let U be the common regular locus of the components g1, . . . , gn ofthe map g:

U := dom (g1) ∩ . . . ∩ dom (gn) ⊂ Rm .

Take a filtration

∅ = E−1 ⊂ E0 ⊂ · · · ⊂ En = K n

for the functions f1, . . . , f p described in Remark 11.6. Then U is the following unionof Zariski locally closed subsets

U =n⋃

i=0

(U ∩ g−1(Ei\Ei−1)

).

Clearly, one of these sets, sayU ∩g−1(Ei0\Ei0−1)must be a Zariski dense open subsetof U and of K m too. Hence f ◦ g is a regular function on U ∩ g−1(Ei0\Ei0−1), whichis the required result. �Corollary 11.8 The zero set Z( f ) of a regulous function f on K n is a closed (in theK -topology) constructible subset of K n.

Corollary 11.9 The zero set Z( f1, . . . , f p) of finitely many regulous functionsf1, . . . , f p on K n is a closed (in the K -topology) constructible subset of K n.

Proof This follows directly from Corollary 11.8 and Lemma 10.1. �Hence and by Proposition 11.1, we immediately obtain

Proposition 11.10 The k-regulous topology on K n is noetherian.

Corollary 11.11 Every k-regulous closed subset of K n is the zero set Z ( f ) of a k-regulous function f on K n and thus is a closed (in the K -topology) constructiblesubset of K n. Hence every k-regulous open subset of K n is of the form

D ( f ) := K n\Z ( f ) = {x ∈ K n : f (x) �= 0}

486 K. J. Nowak

for a k-regulous function f on K n.

Corollaries 11.11 and 11.7 yield the following

Corollary 11.12 Every k-regulous map f : K n → K m is continuous in the k-regulous topology.

12 Regulous Nullstellensatz

We further assume that the ground field K is a Henselian rank one valued field ofequicharacteristic zero and that K is not algebraically closed. Throughout this section,k will be a nonnegative integer. We begin with the following consequence of theŁojasiewicz inequality (Proposition 9.2).

Proposition 12.1 Let f, g be rational functions on KAn such that f extends to a

continuous function on K n and g extends to a continuous function on the set D ( f ).Then the function f s g extends, for s � 0, by zero through the setZ ( f ) to a continuousrational function on K n.

Proof We can find a finite composite σ : M → KAn of blowups along smooth centers

such that the pullbacks

f σ := f ◦ σ and gσ := g ◦ σ

are regular functions at all K -points on

M(K ) and M(K )\σ−1(Z ( f )),

respectively. Then there are regular functions p, q on M such that

gσ = p

qand Z (q) := {y ∈ M(K ) : q(y) = 0} ⊂ Z ( f σ ).

It follows immediately from Proposition 9.2 that the rational function

( f σ )s

q, for s � 0,

extends by zero through the set Z ( f σ ) to a continuous function on M(K ), whenceso does the rational function ( f σ )s · gσ . By the descent property (Corollary 3.6), thecontinuous function ( f σ )s · gσ descends to a continuous function on K n that vanisheson Z ( f ). This is the required result. �

The two corollaries stated below are counterparts of Lemmata 5.1 and 5.2 from [16],established over the real ground field R.


Corollary 12.2 Let f be a k-regulous function on K n and g a k-regulous function onthe open subset D ( f ). Then the function f s g, for s � 0, extends by zero through thezero set Z ( f ) to a k-regulous function on K n. Hence the ring of k-regulous functionson D ( f ) is the localization Rk(K n) f .

Proof The case k = 0 is just Proposition 12.1. Now take s large enough so that thepartial derivatives

f s · ∂ |α|g∂xα

for α ∈ Nn, |α| = 0, 1, . . . , k,

extend to continuous functions on K n vanishing on Z ( f ). Then, by Leibniz’s rule,f s+k g is a k-regulous function on K n and k-flat on Z ( f ), as desired. �Corollary 12.3 Let U be a k-regulous open subset of K n, f a k-regulous function onU and g a k-regulous function on the open subset D ( f ) ⊂ U. Then the function f s gextends, for s � 0, by zero through the zero set Z ( f ) ⊂ U to a k-regulous functionon U.

Proof By Corollary 11.11, U = D(h) for a k-regulous function on K n . From theabove corollary, we get

f hs ∈ Rk(K n) for s � 0.

Hence

g ∈ Rk(K n) f hs

for integers s large enough, and thus the conclusion follows. �Nowwe can readily pass to a regulous version of Nullstellensatz, whose proof relies

on Corollary 12.2 and the fact that the k-regulous topology is noetherian.

Theorem 12.4 If I is an ideal in the ring Rk(K n) of k-regulous functions on K n,then

Rad (I ) = I (Z (I )),

where

I (E) := { f ∈ Rk(K n) : f (x) = 0 for all x ∈ E}

for a subset E of K n.

Proof The inclusion Rad (I ) ⊂ I (Z (I )) is obvious. For the converse one, applyCorollary 11.11 which says that there is a function g ∈ I such that Z (I ) = Z (g).

488 K. J. Nowak

Then Z (g) ⊂ Z ( f ) for any f ∈ I (Z (I )), and thus the function 1/g is k-regulouson the set D ( f ). By Corollary 12.2, we get

f s

g∈ Rk(K n)

for s � 0 large enough. Hence

f s ∈ g · Rk(K n) ⊂ I,

concluding the proof. �

Corollary 12.5 There is a one-to-one correspondence between the radical ideals ofthe ring Rk(K n) and the k-regulous closed subsets of K n. Consequently, the primeideals of Rk(K n) correspond to the irreducible k-regulous closed subsets of K n, andthe maximal ideals m of Rk(K n) correspond to the points x of K n so that we get thebijection

K n � x −→ mx := { f ∈ Rk(K n) : f (x) = 0} ∈ Max(Rk(K n)

).

The resulting embedding

ι : K n � x −→ mx ∈ Spec(Rk(K n)

)

is continuous in the k-regulous and Zariski topologies. Furthermore, ι induces a canon-ical one-to-one correspondence between the k-regulous closed subsets of K n andthe Zariski closed subsets of Spec

(Rk(K n)). More precisely, for every k-regulous

closed subset V of K n there is a unique Zariski closed subset V of Rk(K n) such thatV = ι−1(V ); actually V is the Zariski closure of the image ι(V ).

Proof The embedding ι is continuous by the very definition of the Zariski topology.The last assertion follows immediately from the Nullstellensatz and the fact that theclosed subsets of Spec

(Rk(K n))are precisely of the form

{p ∈ Spec(Rk(K n)

): p ⊃ I },

where I runs over all radical ideals of Spec(Rk(K n)

). �

The above corollary alongwith Proposition 11.10 andCorollary 11.11 yields imme-diately

Corollary 12.6 With the above notation, the space Spec(Rk(K n)

)with the Zariski

topology is noetherian, and the embedding ι induces a one-to-one correspondence


between the k-regulous open subsets of K n and the subsets of Spec(Rk(K n)

)open in

the Zariski topology. In particular, every open subset of Spec(Rk(K n)

)is of the form

U( f ) := {p ∈ Spec(Rk(K n)

): f /∈ p}, f ∈ Rk(K n),

corresponding to the subset D( f ) of K n.

Remark 12.7 As demonstrated in [16] (see also [33, Ex. 6.11]), the ring Rk(K n) isnot noetherian for all k, n ∈ N, n ≥ 2.

From Theorem 12.4 and Corollary 11.11, we immediately obtain

Corollary 12.8 Every radical ideal of Rk(K n) is the radical of a principal ideal ofRk(K n).

Finally, we return to the comparison of the regulous and constructible topologies.Below we state the non-archimedean version of [16, Theorem 6.4] by Fichou–Huisman–Mangolte–Monnier, which says that those topologies coincide in the realalgebraic geometry. The proof relies on their Lemmata 5.1 and 5.2, and it can berepeated verbatim in the case of the ground fields K studied in our paper by means ofCorollaries 12.2 and 12.3.

Proposition 12.9 The k-regulous closed subsets of K n are precisely the closed (in theK -topology) constructible subsets of K n.

The above theorem along with Corollary 11.2 and Corollary 12.5 yields the fol-lowing

Corollary 12.10 There are one-to-one correspondences between the prime ideals ofthe ringRk(K n), the irreducible closed constructible subsets of K n and the irreducibleZariski closed subsets of K n.

Corollary 12.11 The dimension of the topological space K n with the regulous topol-ogy and the Krull dimension of the ring Rk(K n) is n.

13 Quasi-coherent regulous sheaves

The concepts of quasi-coherent k-regulous sheaves on K n and k-regulous affine vari-eties, k ∈ N∪{∞}, can be introduced over valued fields studied in this paper, similarlyas by Fichou–Huisman–Mangolte–Monnier [16] over the real ground fieldR. Also, themajority of their results concerning these concepts carry over to the non-archimedeangeometry with similar proofs. For the sake of completeness, we provide an exposi-tion of the theory of quasi-coherent regulous sheaves. Here we shall deal only withk-regulous functions with a nonnegative integer k, because for k = ∞ we encounterthe classical case of regular functions and quasi-coherent algebraic sheaves.

Consider an affine scheme Y = Spec (A) with structure sheaf OY . Any A-moduleM determines a quasi-coherent sheaf M on Y (cf. [21, Chap. II]). The functor M �−→

490 K. J. Nowak

M gives an equivalence of categories between the category of A-modules and thecategory of quasi-coherent OY -modules. Its inverse is the global sections functor

F �−→ H0(Y,F)

(cf. [21, Chap. II, Corollary 5.5]).Denote by Rk the structure sheaf of the affine scheme Spec

(Rk(K n))and by

Rk the sheaf of k-regulous function germs (in the k-regulous topology equal to theconstructible topology) on K n . It follows directly from Corollaries 12.5 and 12.2 thatthe restriction ι−1Rk of Rk to K n coincides with the sheaf Rk ; conversely, ι∗ Rk =Rk .

By a k-regulous sheaf F we mean a sheaf ofRk-modules. Again, it follows imme-diately from Corollaries 12.5 and 12.6 that the functor ι−1 of restriction to K n givesan equivalence of categories between Rk-modules andRk-modules. Its inverse is thedirect image functor ι∗.

We say that F is a quasi-coherent k-regulous sheaf on K n if it is the restrictionto K n of a quasi-coherent Rk-module. Thus the functor ι−1 induces an equivalenceof categories between quasi-coherent Rk-modules and quasi-coherent Rk-modules,whose inverse is the direct image functor ι∗. For any Rk(K n)-module M , we shalldenote by M both the associated sheaf on Spec

(Rk(K n))and its restriction to K n .

This abuse of notation does not lead to confusion.We thus obtain the following versionof Cartan’s theorem A:

Theorem 13.1 The functor M �−→ M gives an equivalence of categories betweenthe category of Rk(K n)-modules and the category of quasi-coherent Rk-modules. Itsinverse is the global sections functor

F �−→ H0(K n,F).

In particular, every quasi-coherent sheaf F is generated by its global sectionsH0(K n,F).

The regulous version of Cartan’s theorem B, stated below, follows directly fromthe version for affine (not necessarily noetherian in view of Remark 12.7) schemes(cf. [20, Theorem 1.3.1]) via the discussed equivalence of categories (being the functorι−1 of restriction to K n).

Theorem 13.2 If F is a quasi-coherent k-regulous sheaf on K n, then

Hi (K n,F) = 0 for all i > 0.

Corollary 13.3 The global sections functor

F �−→ H0(K n,F)

on the category of quasi-coherent k-regulous sheaves on K n is exact.


Let V be a k-regulous closed subset of K n and I(V ) the sheaf of those k-regulousfunction germs on K n that vanish on V . It is a quasi-coherent sheaf of ideals of Rk ,the sheaf Rk/I(V ) has support V and is generated by its global sections (TheoremA); moreover

H0(

K n,Rk/I(V ))

= H0(

K n,Rk)

/H0 (K n,I(V ))

(Theorem B). The subset V inherits the k-regulous topology from K n and constitutes,together with the restrictionRk

V of the sheafRk/I(V ) to V , a locally ringed space ofK -algebras, called an affine k-regulous subvariety of K n . More generally, by an affinek-regulous variety we mean any locally ringed space of K -algebras that is isomorphicto an affine k-regulous subvariety of K n for some n ∈ N.

We can define in the ordinary fashion the category of quasi-coherentRkV -modules.

Each such module extends trivially by zero to a quasi-coherent Rk-module on K n .The sectionsRk

V (V ) of the structure sheafRkV are called k-regulous functions on V . It

follows from Cartan’s theorem B that each k-regulous function on V is the restrictionto V of a k-regulous function on K n . Hence we immediately obtain the following tworesults.

Proposition 13.4 Let W and V be two affine k-regulous subvarieties of K m and K n,respectively. Then the following three conditions are equivalent:

1) f : W → V is a morphism of locally ringed spaces;2) f = ( f1, . . . , fn) : W → K n where f1, . . . , fn are k-regulous functions on W

such that f (W ) ⊂ V ;3) f extends to a k-regulous map K m → K n.

We then call f : W → V a k-regulous map.

Corollary 13.5 Let W , V and X be affine k-regulous subvarieties of K m, K n andK p, respectively. If two maps

g : W → V and f : V → X

are k-regulous, so is its composition f ◦ g.

It is clear that Cartan’s theorems remain valid for quasi-coherent k-regulous sheaveson affine k-regulous varieties V .

Corollary 13.6 The functor M �−→ M gives an equivalence of categories betweenthe category of Rk

V (V )-modules and the category of quasi-coherent RkV -modules. Its

inverse is the global sections functor

F �−→ H0(V,F).

In particular, every quasi-coherent sheaf F is generated by its global sectionsH0(V,F).

492 K. J. Nowak

Corollary 13.7 If F is a quasi-coherent k-regulous sheaf on V , then

Hi (V,F) = 0 for all i > 0.

Corollary 13.8 The global sections functor

F �−→ H0(V,F)

on the category of quasi-coherent k-regulous sheaves on V is exact.

Note that every non-empty k-regulous open subset U of K n is an affine k-regulousvariety. Indeed, if U = D ( f ) for a k-regulous function f on K n (Corollary 11.11),then U is isomorphic to the affine k-regulous subvariety

V := Z (y f (x) − 1) ⊂ K nx × Ky .

Remark 13.9 Consider a smooth algebraic subvariety X of the affine space KAn . We

may look at the set V := X (K ) of its K -points both as an algebraic variety X (K ) andas a k-regulous subvariety V of K n . Every function f : V → K that is k-regulouson V in the second sense remains, of course, k-regulous on X (K ) in the sense of thedefinition from the beginning of Sect. 11.

Open problem. The problem whether the converse implication is true for k > 0 isunsolved as yet.

For k = 0 the answer is in the affirmative and follows immediately from Theo-rem 10.2. Indeed, every continuous hereditarily rational function f on X (K ) extendsto a continuous rational function on K n , whence f is regulous on V . This theoremwas proven for real and p-adic varieties in [27].

Finally, we wish to give a criterion for a continuous function to be regulous. It relieson Theorem 10.2 on extending continuous hereditarily rational functions on algebraicK -varieties.

Proposition 13.10 Let V be an affine regulous subvariety of K n and f : V → K afunction continuous in the K -topology. Then a necessary and sufficient condition forf to be a regulous function is the following:

(*) For every Zariski closed subset Z of K n, there exist a Zariski dense open subsetU of the Zariski closure of V ∩ Z in K n and a regular function g on U such that

f (x) = g(x) for all x ∈ V ∩ Z ∩ U.

Proof By Corollary 11.5, the necessary condition is clear, because f is the restrictionto V of a regulous function on K n (Corollary 13.3 to Cartan’s theorem B).

In order to prove the sufficient condition, we proceed with induction with respectto the dimension d of the set V which is a closed (in the K -topology) constructiblesubset of K n . The case d = 0 is trivial. Assuming the assertion to hold for dimensions


less than d, we shall prove it for d. So suppose V is of dimension d. By Corollary 11.2,the Zariski closure W of V in K n is of dimension d and we have

W 0 := {x ∈ W : W is smooth of dimension d at x} ⊂ V .

Obviously, W ∗ := W\W 0 is a Zariski closed subset of K n of dimension less than d.Therefore, Y := V ∩ W ∗ is a regulous closed subset of V of dimension less than dand

W = W 0 ∪ W ∗ ⊂ V ∪ W ∗ ⊂ W.

Since the restriction f |Y satisfies condition (*), it is a regulous function on Y by theinduction hypothesis. It is thus the restriction to Y of a regulous function F on K n

(Corollary 13.3 to Cartan’s theorem B).Further, the function f and the restriction F |W ∗ can be glued to a function

g : W = V ∪ W ∗ → K , g(x) ={

f (x) : x ∈ VF(x) : x ∈ W ∗

which satisfies condition (*) as well. Now, it follows fromTheorem 10.2 that g extendsto a regulous function G on K n . Since f is the restriction to V of the function G whichis regulous, so is f , as required. �Acknowledgements The author wishes to express his gratitude to János Kollár and Wojciech Kucharz forseveral stimulating discussions on the topics of this paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate if changes were made.

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